Abstract
Using the theory of weighted Sobolev spaces with variable exponent and the L1-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.
Highlights
Consider the nonhomogeneous and nonlinear Dirichlet boundary value problem: ( )−div(a ( x,u,∇u=)) μ in Ω= u 0 on ∂Ω, where Ω is a bounded open domain of IRN ( N ≥ 2 ) andAu = −div (a ( x,u,∇u)) is a Leray-Lions operator defined from the weighted So-( ) bolev spaces with variable exponent ( ) W01, p(x) Ω,ν into its dual W −1, p′(x) Ω,ν ∗with ν ∗ = ν 1− p′(x) and p(x) + p′( x)= 1
Using the theory of weighted Sobolev spaces with variable exponent and the L1-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure
The study of differential equations with variable exponents has been a very active field in recent years, we find applications in electro-rheological fluids and in image processing
Summary
Consider the nonhomogeneous and nonlinear Dirichlet boundary value problem:. ( ) bolev spaces with variable exponent ( ) W01, p(x) Ω,ν into its dual W −1, p′(x) Ω,ν ∗. ( ) bolev spaces with variable exponent ( ) W01, p(x) Ω,ν into its dual W −1, p′(x) Ω,ν ∗. The problem ( ) is studied where the following assumptions are satisfied:. A typical example of the problem ( ) is the following involving the so-called p ( x) -Laplacian operator with weight:. Under our assumptions (in particular (1.5), the problem ( ) does not admit, in general, a weak solution since the term a ( x,u,∇u) may not belong to ( ) L1loc (Ω). This paper is divided into three sections, organized as follows: In Section 2, we introduce and prove some properties of the weighted Sobolev spaces with variable exponent and, we prove the existence of T -ν -p ( x) -solutions of our problem ( ). Among the research objectives of this article is to introduce it for applications in physics and will be a platform for the problem systems of Dirichlet and others
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